Consider the polynomials f(t) = t + 1 and g(t) = (t + 2)(t + k), where k is an arbitrary constant. For which values of the constant k are the three polynomials f(t), tf(t) and g(t) a basis of P2?

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- Oct 18th 2009, 05:29 PMnoles2188matrix of a linear transformation
Consider the polynomials f(t) = t + 1 and g(t) = (t + 2)(t + k), where k is an arbitrary constant. For which values of the constant k are the three polynomials f(t), tf(t) and g(t) a basis of P2?

- Oct 18th 2009, 05:46 PMBruno J.
This will happen when the matrix of coefficients

$\displaystyle \left(

\begin{array}{lll}

1 & 0 & 2 k \\

1 & 1 & k+2 \\

0 & 1 & 1

\end{array}

\right)$

is nonsingular. Evaluate its determinant, and find the values of $\displaystyle k$ for which it is nonzero.